First of all, I'd like to point out that in some ways this debate is moot: polygamy in some form is already legal in many countries, sex with a close relative is legal in some, and marriage to your half-sibling is legal in Sweden. Any push for more legal recognition in other countries would just be a continuation of what already exists.
I'm honestly unsure about this because there might not be enough demand to legalize either of these two things.
Actually, there's demand for both, because there are people in either type of relationship (sometimes in both at once), and their relationships are extremely illegal in many places. In some states simply cohabitating with multiple people of the opposite gender counts as a common-law marriage, and can get all of you sent to jail for up to 10 years. In most Southern states, as well as some states in Australia, a single sex act with a consenting adult you are related to, regardless of whether you two grew up together, can get both sent to jail for life. There is also a growing community of allies, many from the LGBT+ community, who support relationship and marriage freedom for all consenting adults, regardless of any qualifiers.
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There are also other people publicly speaking out against discrimination and legal abuse. Some are even trying to change culture through positive media representations.
They're real people, and under the current legal regime and set of taboos they suffer, sometimes brutally. Just because you don't personally know of them doesn't mean they aren't there. In fact, you may know some and just don't know you know them. Consanguineous couples especially, are deep, deep in the closet.
I'd also like to point out that according to surveys (and some testimony from people I know), as much as 10% (possibly more) of the population in the US has at least sexually experimented with a sibling (non-coercively, of course). Anyone who advocates for "incest" laws is advocating for throwing as much as a tenth of the entire population into jail.
If we could overcome the genetic problems then would incest be accepted? Not advocating for it definitely, because incidence of birth defect is very high (somewhere above 50% I forget and we are talking serious die before you're 5 years old birth defects and major retardation).
First of all, genetic testing and family planning does allow people to overcome "the genetic problems". Sometimes there are no genetic problems. Your statistic is also incredibly high, and smacks of stereotypes and not science. Of all the reasonably well done studies I've seen, the most pessimistic estimate for genetic problems for 1st degree relatives is 30%. The more realistic estimate I've seen is around 11%, 1.5% lower than for middle aged women. (That's for a single generation, of course.) Those are population statistics anyway. Individual couples may be more or less likely to have problems, just like the general population. Also, there are people in the general population who are not closely related, but both possess terrible genetic diseases which they can pass on to their children, but we don't forbid them from getting married or throw them in jail for having sex. Scientists who know anything about this subject think the eugenic argument is bunk. (I personally find eugenic reasoning to be anti-democratic. It also allows a return to eugenics based on things like race and class). For one, even for the best studies, everyone is aware that their samples are so small and so biased that they can't really even conduct a statistically reliable study beyond 1st cousins, since in most places closer unions are illegal.
"Major retardation" is not the most common defect anyway. The usual problems are related to the immune system or to basic organ function, like the heart. Most of these children are totally fine, and the ones that aren't are still mentally normal. Who is anyone to determine that their lives have no value to themselves or society, and should be prevented at all costs, even at the cost of undermining human rights? Besides, not all consanguineous relationships are even heterosexual.
They are stupid laws. You should be able to do whatever you want with other consenting adults. Marriage is just a contract as well, if you want to marry out of love in a straight or gay relationship, or marry for money, or to get citizenship or because you made a bet, all those should be consider valid reasons. The government shouldn't be deciding who should or shouldn't get married, they should only be recognizing all marriages people ask them to recognize.
Amen! I'd also like to point out that many people don't feel disgust anymore towards the idea of homosexuality is because it's seen frequently in public. Many people still do feel disgust at the idea, even when they support same-sex marriage, and many more felt disgust at the thought in the past. It's a known mechanism, in which seeing another person do something you wouldn't do makes you imagine yourself doing it. The revulsion felt at that mental imagery causes people to lash out at the people who inspired the thoughts. It's the same social psychological mechanism, though perhaps for consanguineous sex the revulsion is stronger for some people.
"Naturalness" also isn't a good rubric for these things. While most animals avoid sex with close relatives in nature, not all do. While most animals have sex with an animal of the opposite sex, not all do. Polygyny, polyandry, and polygynandry all occur in nature. (Polygyny is much more common than monogamy in nature. Should monogamous marriage be illegal?) Which is natural, the majority, or the minority, given that both occur in nature? Besides, neither homophobia, nor the "incest" taboo, occur in nature outside of human culture. Same-sex and consanguineous sex acts are more "natural" by many standards, than the taboos and laws against them.
Sexual morality is too important to the happiness and the well-being of us all to be determined by superstition, politics, economics, and religious taboos.